Thursday, January 10, 2013

Sharing the peanut butter sandwich

The other day, a great blogger named Christopher Danielson, who thinks about units and place value on a level that I hope to reach some day, posed an intriguing question:

I like this question because it asks us to get at the heart of how kids (and adults untainted by math instruction) conceptualize numbers. I responded that I thought 1 1/2 made more sense because 3 sixths would require an implicit cutting of all of the pieces of what I interpreted to be the quesadilla (because, really, who splits a cookie into thirds? Who shares cookies at all?) into six pieces, in order to share them equally. I applied what I thought Barbara and Sebastian would do, which is to take one piece each, then split the remaining piece in half.

I asked Barbara and Sebastian this question, using a sheet of paper I'd cut into thirds, and presented it like this:

I'm going to take this paper and cut it into three equal-sized pieces, and then you and I are each going to take the same amount of paper. This is called 1/3, this is called 1/3, and this is called 1/3.

Barbara gave herself two of the pieces and gave me one. She was done.

I made a face and told her this wasn't fair. She switched so I had 2/3 and she had 1/3. I made another face and she said, "or, we could rip this one." She ripped the third piece, so now we each had one of the original 1/3 pieces and half of the remaining 1/3 piece. I asked her how much she had and she said I have one piece, and a smaller piece.

I asked her what the smaller piece was called and she was stumped. "A little piece." I reported that this was the answer our kids gave, committing the sin of assuming understanding based on the most advanced student's response.

I tried the same thing with Sebastian today. Though there are some highlights, my main takeaway was that I wanted to have a conversation about math, making it relevant by using the fact that he'd just asked me for a peanut butter sandwich, but he really just wanted a peanut butter sandwich. When talking math with kids, timing is everything:

Some thoughts:

1. Both of our kids first thought that the best way to share was to take 2/3 for themselves and leave me 1/3. This is either an indictment or a triumph of our parenting style.

2. This moment with Sebastian, feeling silly and hungry, reminded me that often, when we think we're talking math with our kids, they are really more into the surface structure of the task. The underlying work with number sense, mathematical problem solving, and struggle to name fractional amounts take up no more than 25% of Sebastian's working memory, the rest of it filled with 5-year-old versions of ideas like "why can't you just make your own damn sandwich?" and "now can I eat it?"

3. After thinking about this for the last two days, and trying this with both Barbara and Sebastian, I am struck by the ease with which each of them decided to split the remaining piece, and the lack of vocabulary to describe the precise amount that each piece represented. I was also struck that, by essentially telling them that each of the three pieces was 1/3 of the whole, I was naming a new unit for them, giving a name to what they otherwise would call a "piece." I don't know enough about early childhood development to describe this precisely, but they are at a stage where they count things. I could have called each third a whatsit, and they would have described the final amount the same way: A whatsit and a smaller piece. The term 1/3 means nothing as of right now, but the concept of splitting things into equal-sized parts makes a lot of sense. Not in some abstract sense, but in a very real, rip-the-bread kind of way.

4. Another take-away for me was that the reason we wouldn't accept either kid's answer beyond a certain age is that, at least in math classes, we expect every part of an amount to be expressed in terms of a single unit. If the unit is the whole sandwich (or quesadilla, or piece of paper), we want to know how much of it we have. If the unit is a third, we want to know how many thirds. We don't want to know how many thirds and how many other pieces, nor do we ask for the number of thirds and fourths. Egyptians didn't write fractions this way back in the day, expressing fractions as the sum of unit fractions, but I'm not taking this as evidence that they actually perceived fractions that way. I have a hard time believing that the pharaohs really conceived of Sebastian's original partition (2/3) as 1/3 + 1/4 + 1/12, meaning "OK, let's take one of these original pieces, mark it for later, put the original back together and cut into four pieces, take one of those and mark it, then put the original back together and cut into twelve pieces, mark it, and take the marked pieces. No, wait; that's not even. I don't know how they viewed fractions conceptually, but I know that our current representation differs from how both of my kids (small sample size, grew up in the same home, but still) view fractions, and I imagine this concept/notation gap is not unique to our household or culture.

The same-unit idea is a new one for me, though. It got me to thinking that we don't say things like "the pencil was five inches and three centimeters long", but we can say, "the table was three feet, six inches tall" because inches and feet share a common unit (the unit either being the inch, decomposed from feet, or the foot, composed from inches.) For what it's worth, I think "three and a half feet" very clearly refers to the foot as the unit, whereas "three feet, six inches" uses the inch as the basic unit. But at some point, kids realize that an answer to "how much of the quesadilla do you have?" should not be answered with "one piece of this size, and one piece of this unrelated size."

5. Perhaps most importantly, I need to spend more time talking math with my kids. Not because they need to get ahead, or because of increasing global competition, but because it lets the people I love the most fall in love with the subject I love the most, and makes us all think in the process.


  1. And to think I was proud of counting change with my kindy child and telling her the values of each coin. You'r already introducing fractions! It's interesting to me that your love of math translates to what you want your children to understand and love. I liked math ok but anything english or literature related was my true passion. And imagine my horror to only have 1 child who reads for pleasure. We have loads and loads of books on many bookshelves here but the children don't ever browse our shelves to pick a book. I loved doing that as a child. I have to pick out books for them and encourage them to read. Anyhow, glad to hear you are WRITING...cause I kind of like that sort of thing.

  2. Mrs. Halligan - you should be very proud of counting change, encouraging reading, and everything else you're doing as a parent. You have kids that care about others, and who have developed a positive attitude about the world. For what it's worth, as a child I really hated reading (despite the repeated insistence of my parents and teachers that I read), and now it's one of my favorite things to do. Sometimes the benefits come much later - just keep encouraging them to do things you love, be genuinely enthusiastic about it, and they'll pick up on the joy of loving something worthwhile.

  3. Rob, this is really interesting. I frequently find that my tutoring students (middle and high school, small sample size, same district) have a disproportionate (no pun intended) difficulty with manipulating and dealing with fractions - at all levels of Math. How much of this do you think stems from a disconnect between our natural understanding (1 and 1/2 thirds) and our computational understanding (1/3*1/2 or -even more unintuitive -(1/3)/2)? Is it a case where we have to re-train natural intuition to create, as I've been calling it with my students, 'Mathematical intuition', or is there a way to harness our '1 and 1/2 thirds' intuition to handle more complex concepts like thinking of parts, wholes, and anything in between?

  4. Hey Alex, good to see you here. It sounds like your hypothesis is that 1) language is clunky, which causes dissonance between what we intuit and how we've been taught to answer, and 2) many students (or at least some) are so used to thinking of math as a series of correctly worded numerical answers that they haven't developed (or have forgotten) a way to make solid sense of what is going on.

    If this is what you're saying, I agree. And I think the solution is to pose the problem as a fun challenge, have your students make sense of the situation through concrete and pictorial representations, and then talk about different ways to express the work they've done. By the time you get to "one and a half thirds", you can remind them that fractions need to be expressed in terms of a single sized piece. So this is a perfectly fine way to talk about it, but when you're going to answer questions in class, you need to boil it down to how many (a whole number) of a single-sized piece...then you show how to cut the thirds into sixths to express this as 3/6.

    The key is to let the words flow without being impeded by norms at first, and to get back to our natural ability to understand what it means to share. Only then is it a good idea to talk about standard notation for fractions.

    If you try this, let me know how it works out for you. Best of luck!